And the browser would then actively listen for a message from that originator (which could either arrive directly on the device or, via something like Apple's Messages app, a message sent to your phone could auto-fill a field on your mac).

So when the following SMS arrives;

Your one time passcode is [912341851]

The user is prompted to auto-fill the form with the passcode provided.

Likewise this would make for easy phone number verification for apps like WhatsApp and such like, rather than requiring the user to exit the app, read the code, and return to the app.